How do you program fear into a neural network?
There are a lot of approaches you could take for this. Creating a realistic artificial analog for fear as implemented biologically in animals might be possible, but there is quite a lot involved in a real animal's fear response that would not apply in simpler AI bots available now. For instance, an animal entering a state of fear will typically use hormones to signal changes throughout its body, favouring resource expenditure and risk taking ("fight or flight").

In basic reinforcement learning, the neural network would not need to directly decide switch on a "fear mode". Instead, you can make use of some design in the agent and learning algorithm to help learn from rare but significant events. Here are a few ideas:


Experience replay. You may already be doing this in the Pacman scenario, if you are using DQN or something similar. Storing the state transition and reward that caused a large positive or negative reward, and repeatedly learning from it should offset your concern
Prioritised sweeping. You can use larger differences experienced between predicted and actual reward to bias sampling from your replay memory towards significant events and those linked closely to them. 
Planning. With a predictive model - maybe based on sampled transitions (you can re-use the experience replay memory for this), or maybe a trained state transition prediction network - then you can look multiple steps ahead by simulating. There is a strong relation between RL and look-ahead planning too, they are very similar algorithm. The difference is which states and actions are being considered, and whether they are being simulated or experienced. Experience replay blurs the line here - it can be framed as learning from memory, or improving predictions for planning. Planning helps by optimising decisions without needing to repeat experiences as much - a combination of planning and learning can be far more powerful than either in isolation.
Smarter exploratory action selection. Epsilon-greedy, where you either take a greedy action or take a completely random action, completely ignores how much you may have already learned about alternative actions and their relative merit. You can use something like Upper Confidence Bound with a value-based agent.
In a deterministic world, increase the batch size for learning and planning, as you can trust that when a transition is learned once, that you know everything about it. 


You will need to experiment in each environment. You can make learning agents that are more conservative about exploring near low reward areas. However, if the environment is such that it is necessary to take risks in order to get to the best rewards (which is often the case in games) then it may not be optimal in terms of learning time to have a "timid" agent. For instance in your example of Pacman, sometimes the ghosts should be avoided, sometimes they should be chased. If the agent learned strong aversion initially, it might take a long time to overcome this and learn to chase them after eating a power-up.

For your example of the spider, as the constructor of the experiment then you know that the bite is bad every time and that the agent must avoid it as much as possible. To most RL algorithms, there is no such knowledge, except gained through experience. An MDP world model does not need to match common sense, it may be that a spider bite is bad (-10 reward) 90% of the time and good 10% of the time (+1000 reward). The agent can only discover this by being bitten multiple times . . . RL typically does not start with any system to make assumptions about this sort of thing, and it is impossible to come up with a general rule about all possible MDPs. Instead, for a basic RL system, you can consider modifying hyperparameters or focusing on key events as suggested above. Outside of a basic RL system there could be merit in replicating other things, such as "instinctive" fear.